Bottom Line:
It generally remains unclear, however, how this mutual inactivation and the resulting switching behavior can impact developmental patterning circuits.For lateral inhibition, we find that mutual inactivation speeds up patterning dynamics, relieves the need for cooperative regulatory interactions, and expands the range of parameter values that permit pattern formation, compared to canonical models.Together, these results provide a framework for analysis of more complex Notch-dependent developmental systems.

pcbi-1002069-g003: Boundary width is robust to correlated noise in Notch and Delta.(A) Notch reporter profiles (green heat map, bottom panel) for varying maximal production rates of DSL, (red curves in top panel) and a fixed production rate of Notch, (blueline). Spatially-uniform reduction in levels (y-axis, lower panel) results in restriction of the vein to a progressively narrower region (lower panel). However, when the DSL production rate is lowered to the extreme when everywhere, all cells are in receiver states, and vein boundaries are no longer restricted by Notch signaling (see discussion in text). This is the expected behavior in the Delta+/− heterozygous mutant when the DSL production rate is half that of the wild-type (arrows), if in the wild type . (B) The Notch reporter profile is sensitive to intrinsic (uncorrelated) noise but robust to extrinsic (correlated) noise in Notch and DSL production rates. Simulations of boundary formation with static multiplicative production rate noise of similar magnitude but different degrees of correlation (blue scatter plots) show that the pattern is less sensitive to extrinsic noise (top) than intrinsic noise (bottom). (C) The effect of noise amplitude and degree of correlation on Notch reporter peak positions. Standard deviation in peak position (color bar) at each row (red dots in B) is calculated from 300 simulations of 8×24 cell arrays (such as those in B) for different noise attributes. The noise parameters used in B are marked (white circles). See Supporting Information Text S1.5 and Table S1 for parameter values and description of noise generation.

Mentions:
A striking aspect of the Drosophila wing vein system is observed in the heterozygous mutants of Notch and Delta (e.g. single copies of the Notch and Delta genes). While heterozygous mutants of Notch (Notch+/−) or Delta (Delta+/−) alone exhibit mutant phenotypes (causing thicker veins), the Notch+/− Delta+/− double mutant restores the wild-type phenotype [19], [20], [21]. More generally, several mutant phenotypes seem to depend on the ratio between the copy numbers of the Notch and DSL genes [19]. This ratiometric dependence of the vein width cannot be derived from the several known feedbacks operating in the Drosophila wing vein, but emerges automatically from the MI model. This is because the position of the Notch signaling band occurs where Notch and DSL production rates are equal. This position remains unchanged when both rates are multiplied by the same factor. By the same reasoning, the vein width (distance between side bands) increases with increasing ratios between the effective copy numbers of DSL and Notch, as shown in Figs. 3A, S2.

pcbi-1002069-g003: Boundary width is robust to correlated noise in Notch and Delta.(A) Notch reporter profiles (green heat map, bottom panel) for varying maximal production rates of DSL, (red curves in top panel) and a fixed production rate of Notch, (blueline). Spatially-uniform reduction in levels (y-axis, lower panel) results in restriction of the vein to a progressively narrower region (lower panel). However, when the DSL production rate is lowered to the extreme when everywhere, all cells are in receiver states, and vein boundaries are no longer restricted by Notch signaling (see discussion in text). This is the expected behavior in the Delta+/− heterozygous mutant when the DSL production rate is half that of the wild-type (arrows), if in the wild type . (B) The Notch reporter profile is sensitive to intrinsic (uncorrelated) noise but robust to extrinsic (correlated) noise in Notch and DSL production rates. Simulations of boundary formation with static multiplicative production rate noise of similar magnitude but different degrees of correlation (blue scatter plots) show that the pattern is less sensitive to extrinsic noise (top) than intrinsic noise (bottom). (C) The effect of noise amplitude and degree of correlation on Notch reporter peak positions. Standard deviation in peak position (color bar) at each row (red dots in B) is calculated from 300 simulations of 8×24 cell arrays (such as those in B) for different noise attributes. The noise parameters used in B are marked (white circles). See Supporting Information Text S1.5 and Table S1 for parameter values and description of noise generation.

Mentions:
A striking aspect of the Drosophila wing vein system is observed in the heterozygous mutants of Notch and Delta (e.g. single copies of the Notch and Delta genes). While heterozygous mutants of Notch (Notch+/−) or Delta (Delta+/−) alone exhibit mutant phenotypes (causing thicker veins), the Notch+/− Delta+/− double mutant restores the wild-type phenotype [19], [20], [21]. More generally, several mutant phenotypes seem to depend on the ratio between the copy numbers of the Notch and DSL genes [19]. This ratiometric dependence of the vein width cannot be derived from the several known feedbacks operating in the Drosophila wing vein, but emerges automatically from the MI model. This is because the position of the Notch signaling band occurs where Notch and DSL production rates are equal. This position remains unchanged when both rates are multiplied by the same factor. By the same reasoning, the vein width (distance between side bands) increases with increasing ratios between the effective copy numbers of DSL and Notch, as shown in Figs. 3A, S2.

Bottom Line:
It generally remains unclear, however, how this mutual inactivation and the resulting switching behavior can impact developmental patterning circuits.For lateral inhibition, we find that mutual inactivation speeds up patterning dynamics, relieves the need for cooperative regulatory interactions, and expands the range of parameter values that permit pattern formation, compared to canonical models.Together, these results provide a framework for analysis of more complex Notch-dependent developmental systems.